Quantum State Vectors: Algebraic Operations

The binary system that is used in digital devices has two states: 0 and 1. In quantum computing, these states are represented as vectors; but unlike normal vectors, state vectors are denoted using the ket notation, |v⟩. Thus, 0 = |0⟩ and 1 = |1⟩. Quantum state vectors simply point to a specific point in space that corresponds to a particular quantum state.

https://www.ibm.com/quantum-computing/

A vector space, V, over a field, F, is a set of vectors where two conditions hold:

1. Adding two vectors that are elements of the vector space, V, will yield a third vector, which is also an element of V.

2. Scalar multiplication between a vector,| a⟩ ∈ V, and a scalar, n ∈ F, is also a member of V.

Algebraic Properties of Vectors

State vectors can be expressed in the [x,y] format such that:

State Vector | 0⟩

State Vector | 1⟩

Recap:

Vector Addition — The sum of two vectors is another vector.

Given vectors |α⟩and|β⟩: |α⟩+|β⟩ = |γ. In normal vector operations:

Vector Addition

Additive Properties of Vectors

1. Vector addition is commutative. |α⟩+|β⟩ = |β⟩ +|α⟩

2. Vector addition is associative. |α⟩+(|β⟩+|γ⟩)= (|α⟩ +|β⟩)+|γ⟩

The null/zero vector, |0⟩, has two properties that are shown in equations 1 and 2 below.

There exist a null vector with the property |α⟩ + |0⟩ = |α⟩

For every vector |α⟩, there exist an inverse vector (1-|α⟩) such that |α⟩ + (1-|α⟩) = |0⟩

Vector Multiplication — vector multiplication can be done in three ways: scalar, dot, or cross product.

Scalar Product involves the multiplication of a vector by some real number m or n.

1. Commutative Law of Multiplication

Normal Vector Addition operation

Commutative Law of Multiplication, where m is a scalar

2. Associative Law of Multiplication

Associative Law of Multiplication where m and n are scalars

Associative Law on normal vectors

2. Distributive Law of Multiplication

Distributive Law

Distributive Law

Matrix Multiplication

Matrices are mathematical objects that transform vectors to other vectors:

| v⟩ → | v′⟩ = M |v⟩. Matrix multiplication is given by the general formula:

https://mathinsight.org/matrix_vector_multiplication#:~:text=We%20define%20the%20matrix%2Dvector,m%C3%971%20column%20vector.

Vector multiplication can also be an inner/dot product or a cross product.

Inner/Dot Product

In normal vector operations, the dot product of two vectors, A and B, is defined as the product of the magnitudes of A and B and the cosine of the angle θ between them. Therefore, A.B = AB cos θ and A.B is a scalar.

Dot Product

However, the dot product of two state vectors |α⟩and|β⟩ is a complex number:

The dot product has the following properties:

1. Commutative

2. The dot product of a vector by itslef is greater than or equal to 0 and |α⟩.|α⟩ is only equal to zero if |α⟩ is 0.

Property 2 from Griffiths Introduction to Quantum Mechanics

3. Associative

Cross Product

The cross product of A and B is a vector C = AxB (read A cross B), which is the product of the magnitudes of A and B and the sine of the angle θ between them.

AxB = AB sin θ u where is a unit vector indicating the direction of A B.

Properties of the cross product will be discussed in a different article.

Going forward, these algebraic properties will be essential in discussing the characteristics of state vectors and quantum gates.